Speaker
Description
We formulate finite exact renormalization group (ERG) transformations on a multiply connected configuration space $Q$ by using the universal covering map $\pi:\widetilde Q\to Q$ and the Deck transformation group $\Gamma\cong\pi_1(Q,q_0)$. Following the broad definition used by Igarashi, Suzuki, and Sonoda, an ERG transformation is treated as a linear integral transformation acting on Boltzmann factors. In this formulation, a finite ERG transformation is represented by an integral kernel $K_{\Lambda,\Lambda'}$ between two cutoff scales $\Lambda\leq\Lambda'$, rather than by its infinitesimal flow equation. The construction uses Dowker's covering-space method, where kernels on a multiply connected space are obtained from kernels on the universal cover and characters of the fundamental group. Let $e^{\widetilde S_t[\widetilde q]}$ be a cutoff-dependent Boltzmann factor on $\widetilde Q$, transforming under Deck transformations by a character $\chi:\Gamma\to\mathbb C^\times$. If the ERG kernel and the measure are invariant under Deck transformations, then the transformed Boltzmann factor has the same character $\chi$. Thus, the sector labeled by the $\pi_1$-character is preserved under finite ERG transformations. This gives a finite-kernel formulation of the non-renormalization of the $\pi_1$-character. We also derive the Gaussian ERG kernel on the universal cover from the scaling relation for the normal-ordering generating functional and describe the constraint on $\chi$ imposed by normalization on $Q$.
| Affiliation | Nihon University |
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| Career status | PhD student |